Centered trochoid

In geometry, a centered trochoid is the roulette formed by a circle rolling along another circle.

The term encompasses both epitrochoid and hypotrochoid.

Alternatively, a centered trochoid can be defined as the path traced by the sum of two vectors, each moving at a uniform speed in a circle.

Specifically, a centered trochoid is a curve that can be parameterized in the complex plane by or in the Cartesian plane by where If

is rational then the curve is closed and algebraic.

Otherwise the curve winds around the origin an infinite number of times, and is dense in the annulus with outer radius

However, some authors (for example [1] following F. Morley) use "trochoid" to mean a roulette of a circle rolling along another circle, though this is inconsistent with the more common terminology.

The term Centered trochoid as adopted by [2] combines epitrochoid and hypotrochoid into a single concept to streamline mathematical exposition and remains consistent with the existing standard.

A trochoidal curve can be defined as the path traced by the sum of two vectors, each moving at a uniform speed in a circle or in a straight line (but not both moving in a line).

In the parametric equations given above, the curve is an epitrochoid if

be rolled on a circle of radius

is attached to the rolling circle.

The fixed curve can be parameterized as

and the rolling curve can be parameterized as either

depending on whether the parameterization traverses the circle in the same direction or in the opposite direction as the parameterization of the fixed curve.

be attached to the rolling circle at

Then, applying the formula for the roulette, the point traces out a curve given by: This is the parameterization given above with

remains the same if the indexes 1 and 2 are reversed but the resulting values of

This produces the Dual generation theorem which states that, with the exception of the special case discussed below, any centered trochoid can be generated in two essentially different ways as the roulette of a circle rolling on another circle.

The point p is 1 unit from the center of the rolling so it lies on its circumference.

This is the usual definition of the cardioid.

In this case the fixed circle has radius 1, the rolling circle has radius 2, and, since c > 0, the rolling circle revolves around the fixed circle in the fashion of a hula hoop.

This produces an essentially different definition of the same curve.

then we obtain the parametric curve

, this is the equation of an ellipse with axes

This gives two different ways of generating an ellipse, both of which involve a circle rolling inside a circle with twice the diameter.

in both cases and the two ways of generating the curve are the same.

In this case the curve is simply

This degenerate case, in which the curve is a straight-line segment, underlies the Tusi-couple.